The unique ergodicity of equicontinuous laminations

Ergodic decompositions for folding and unfolding paths in Outer space

We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup $H$ of $Out(\FN)$ quasi-isometrically embeds into the complex of free factors v…